Let your inputs go!

As one learns more about feedback and control systems, at some point, comes an important realization that a variety of distinct problems can be answered if we know how to answer the question: “is some closed-loop system asymptotically stable?” Indeed, the study of stability dominates Chapters 6 and 7, arguably the densest chapters in the book. Many students, when asked what they remember from their undergraduate control class, will quickly point at the root-locus or at the Nyquist stability criterion. Yet, in the beginning, it might seem that stability is a side dish and not the main attraction.

Take for example Chapter 4. It is easy to overlook that asymptotic stability is a requirement for tracking. Or to mistake internal stability as something that is only important in the rare case of pole-zero cancellations. It is common for students to wonder what to do with the system inputs when asked to analyze a feedback system using the root-locus method or the Nyquist stability criterion. For example, here is a common question:

How do you incorporate a constant disturbance into your $L$ transfer function? In the case of this block diagram(apologize for the messiness), I was unable to separate an alpha and C(s) from the K(s) since there is a constant omega [$w$] which would be added to the output of e*K(s). So in this case, do we just ignore omega [$w$]?


The above question reveals some apprehension about “ignoring” the inputs, as if one would be answering only the stability question. It also reveals that the student is likely missing a very important point: the reason for reformulating the problem as a stability question is not to ignore the inputs but rather to look for answers that hold for all possible “well behaved” inputs. Stability provides broad guarantees rather than specific answers.

It might be especially difficult to “let the inputs go” if they have physical meaning. Take for example a problem such as P6.34-P6-36, in which students are asked to revisit the design of a controller for the temperature, $T$, of a water heater using the root-locus method. A model for the system is the differential equation

$$\dot{T}(t)+\left (\frac{v}{m} + \frac{1}{m c R} \right ) T(t) = \frac{1}{m c} \left (u(t) + w(t)\right ) $$

in which the “disturbance” is the very physical signal

$$w(t) = \bar{w}, \quad \bar{w} = v \, c \, T_i +\frac{1}{R} T_0,$$

which is related to things like the system’s thermal properties, $c$ and $R$, and the (constant) in/out flow and its temperature, $v$, $T_0$, $T_i$, which clearly make $\bar{w} \neq 0$. In the presence of a model like this, it is not uncommon to hear someone say: “It doesn’t make any sense to analyse this system if the input $w(t)=\bar{w}$ is not present!”

Yet, because stability allows you to draw conclusions about a large class of inputs, namely all bounded inputs, which $w(t) = \bar{w}$ is certainly a member of, it is safe to go ahead and ask whether a controller will stabilize the closed-loop system without worrying about the particular input $w(t) =\bar{w}$.

Likewise, a question such as “does the closed-loop controller track a given constant reference $\bar{y}$?” can be answered by adding a pole at the origin to the controller (see Lemma 4.1) and assessing asymptotic stability of the resulting closed-loop transfer-functions.

Of all the tricks you learn with feedback control this is one of my favorites. So next time, just let your inputs go! Stability will take care of them.

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